Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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33 views

How to compute this integral : $\oint \bar{z}^n dz$

How to compute this integral : $$\oint_{|z|=a} \; \bar{z}\;^n dz$$ I choose $z = ae^{i \theta}$, and so $\bar{z}\;^n = a^n e^{-i\theta}$ And $$\oint_{|z|=a} \; \bar{z}\;^n dz = ...
0
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2answers
63 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } ...
1
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0answers
26 views

Integraion of the function $1/r$ over a sphere in $\mathbb{R}^3$

Assume in $\mathbb{R}^3$ there is a sphere $S=S(A,R)$ centered at a point $A$ with radius $R$, and $|A|=a$, where $|A|$ is the Euclid norm of $A$. Now let a $X$ be a uniform distributed random point ...
1
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1answer
44 views

How to justify, $\sum_{n=1}^{\infty} a_{n} x^{n} - \sum_{n=1}^{\infty}a_{n}y^{n}=\sum_{n=1}^{\infty} a_{n} (x^{n}-y^{n})$?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$ and we let $K_{1}$ be a compact subset of ...
0
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1answer
31 views

What is the formal meaning of “determine” in Baby Rudin 2.40?

In Theorem 2.40, Rudin talks about a $k$-cell $I$ formed by the intervals $[a_1, b_1], \ldots, [a_k, b_k]$. We split each interval at its midpoint $c_j = \frac{a_j + b_j}{2}$ and end up with $2k$ ...
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0answers
24 views

Predictor-Corrector for Adams-Moulton

What is the order of the corrector of Adams-Moulton type required in order to apply Milne's method for estimating the error in PECE mode? Find the coefficient of the leading term in the truncation ...
5
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1answer
70 views

integration in five dimensions space

I am doing this problem: Consider the differential form $$a=p_1 \, dq_1+p_2 \, dq_2-(p_1^2+p_2^2+q_1^2+q_2^2) \, dt\text{ in }\mathbb R^5=(p_1,p_2,q_1,q_2,t).$$ (a) Compute the differential $da$ and ...
3
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3answers
59 views

Analysis on manifolds after course on Lebesgue integration

I am an junior currently taking a course on measure theory and Lebesgue integration using Royden's text. Before this, I took a standard intro to analysis course covering the first seven chapters of ...
3
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1answer
42 views

Is this double integral always positive on nonzero continuous functions?

Is this double integral $$(f,g)=\int_{x=0}^1\int_{y=0}^1\frac{f(x)g(y)}{|y-x|^{\frac14}}dydx$$ an inner product on continuous functions on $[0,1]$? Namely, is $(f,f)$ always positive for all nonzero ...
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0answers
24 views

How to show the inequation of two function on two different values

Denote $v_1(\lambda)=\frac{(1-\lambda)(2-\lambda)-1}{\sqrt{1+(1-\lambda)^2+((1-\lambda)(2-\lambda)-1)^2}}$ and $v_2(\lambda)=\frac{1-\lambda}{\sqrt{2+(1+\lambda)^2}}$. The figure shows the curve of ...
0
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1answer
32 views

Question relating to the Casorati-Weierstrass Theorem.

The question I am trying to answer is: Suppose $f$ is analytic in the punctured disc $0 < |z| < 1$ except for poles $\{z_n\}$ where: $$\lim_{n \to \infty}z_n = 0$$ Note that $0$ is not an ...
1
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1answer
29 views

Limit of sequence that is root of sums of powers

Suppose $z_1,\ldots,z_k$ are complex numbers with $|z_1|>\cdots>|z_k|$, and let $c_1,\ldots,c_k$ be non-zero complex numbers. Let $a_0,a_1,\ldots$ be the sequence defined by $$a_n=\sum_{i=1}^k ...
1
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1answer
42 views

How to setup a sequence of functions?

I have a function $F(x,t)=\int_0^t f(s,x)ds$ and I want to see if I can write $$\frac{\partial F(x,t)}{\partial x}=\int_0^t \frac{\partial f(s,x)}{\partial x}ds$$ So, I basically want to know if I ...
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3answers
69 views

Question on Uniform Conergence

I need to show that $\sum_{k=1}^\infty$$(\frac {x}{2})^k$ does not converge uniformly on (-2, 2) I know I have to show that $\sup_{x\in X}\lvert f_n(x)-f(x)\rvert\nrightarrow0 $ as ...
0
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1answer
26 views

Almost continuity implies measurability?

Trying to prove the continuity of $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ $(n>1)$ I got the following property of $ f $: for all $x\in \mathbb{R}^n $ and $(x_k)$ such that $x_k \rightarrow x$ ...
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0answers
25 views

Construct a Converging Series from the Following

This is more of a request for advice than a request for solution. Last night we were given the following and nobody figured it out in the time given (about 5 minutes). I think this is a problem many ...
1
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1answer
24 views

Prove Uniform Convergence of Series of functions-Help?

Let F0 be a bounded Riemann integrable function on [0, 1]. For n āˆˆ N, define $F_n(x)$ on [0,1] by $F_n(x)$ = $\int_{0}^{x}$ $F_{n-1}(t)$ dt 1) Prove that for all nāˆˆ N and xāˆˆ [0,1], we have ...
0
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1answer
29 views

Can we write $f\in C^{1}(\mathbb R^{2})$ as $f(z_{1})-f(z_{2})= (z_{1}-z_{2})\cdot G(z_{1}, z_{2})$?

Mean-value theorem for one variable, tells us that if $f:\mathbb R \to \mathbb R$ is continuously differentiable, then we can write, $f(x)-f(y) = (x-y) G(x,y)$; where $x,y \in \mathbb R$ and actually ...
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3answers
36 views

$\frac{1}{a_n}\int_0^{a_n} f(x) \,dx \rightarrow f(0)$ if $a_n\rightarrow 0$

The full question I'm looking at: Suppose that $f:[0,1]\rightarrow\mathbb{R}$ is continuous. Suppose $a_n>0$ satisfy $a_n\rightarrow 0$ as $n\rightarrow \infty$. Prove that ...
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0answers
32 views

Question about deformation

Why if $H\setminus X$ has no crititical point then $X$ is a deformation retract of $H$? $H$ is a Hilbert space $X$ is a subspace of $H$ ($X=\varphi^b=\lbrace x\in H, \varphi(x)\leq b\rbrace$ , ...
2
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2answers
44 views

Is it true that, $|e^{x}-e^{y}|\leq C \cdot |x-y|$?

Define $f:\mathbb R \to \mathbb R$ such that $f(x)= e^{x}-1:= \sum_{n=1}^{\infty} \frac{x^{n}}{n!};$ for $x\in \mathbb R.$ My Question: Can we expect $|f(x)-f(y)|\leq |x-y| \cdot C;$ where $C$ is ...
2
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2answers
48 views

Derivatives on Functors

I'm not even sure if this question makes pedantic sense but is there any way to rigorously define the notion of taking the derivative of a functor?
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0answers
33 views

Higher-dimension integrability (over rectangles) well-defined

Here is the problem and my work toward a proof: Question: Prove that in the following definition, the value of $\int_E f dx$ is independent of the choice of rectangle $J$: Definition: ...
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0answers
21 views

A simple proof with directional derivatives

Suppose $\nabla f (x) \cdot d < 0$, prove that there exists $\delta > 0$ such that $$f(x + \tau d ) < f(x)$$ for all $\tau \in (0, \delta)$ My proof consists of only a few lines. ...
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1answer
25 views

Eigenvalues of adjoint for residual spectrum.

Statement: Let $T$ be a bounded operator in a Hilbert space $\mathscr{H}$ Show that if $T-\lambda I$ is not dense in $\mathscr(H)$, then $\overline{\lambda}$ is an eigenvalue of $T^*$. Attempted ...
19
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2answers
255 views

When does $(uv)'=u'v'?$

In any calculus course, one of the first thing we learn is that $(uv)'=u'v+v'u$ rather than the what I've written in the title. This got me wondering: when is this dream product rule true? There are ...
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4answers
42 views

Determine if the given sequence converges or diverges

Let $(x_n)$ be a sequence defined as $x_n = \frac{1}{n} \sum_{j=1}^{n} \frac{j+1}{j^2}$ . We want to know if $(x_n)$ converges. The trouble I am having here is that the sum depends on $n$. We know the ...
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0answers
27 views

Show $f$ is concave up if and only if graph of $f$ is above tangent line at every point

I think that this problem is intuitively obvious, and may involve Jensen's Inequality, but I am not really sure how to prove it. Any help is appreciated! Thanks
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1answer
21 views

Analysis Proof of Inflection Points

We are supposed to prove this, and it seems relatively simple, but as per usual, I don't know where to start. I assume that a big factor is that the third derivative is not zero at $x_0$, which ...
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0answers
14 views

Continuity of Complex function and restrictions

I am trying the following question but am stuck at finding the restriction: Prove that $f(z)=1/z^2$ is continuous at $z_0= 1+2i$ Solution: I am trying the use epsilon-delta proof and got it down to: ...
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4answers
118 views

Does $xy\geq x+y$?

I just see the GM-AM inequality. But I would like to compare $xy$ with $x+y$ for any $(x, y)\in\mathbb{R}^2$. It looks like $xy>x+y$ since the first one is multiplication and the second one is ...
0
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1answer
60 views

Is this construction already a contradiction

Let's say we have $\ell^p$ for $p>2$ and $\ell^2 \subsetneq \ell^p$. Let $U$ be a closed subspace of $\ell^2$. Is it possible that $U$ is isomorphic to $\ell^p$ as a Banach space?- I hardly think ...
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2answers
32 views

Linear operator exists then differentiable?

Let $E_{\text{open}} \subseteq \mathbb{R}^n$, and let $\vec{x_o} \in E$. Let $\vec{f}: E \rightarrow \mathbb{R}^m$. If there exists a linear operator $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$. such ...
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0answers
18 views

Showing uniqueness of character identity

How would one show that any complex-valued C1 function satisfying the character identity must be of the form exp(cx) for c complex. Given a function f, it is said to satisfy the character identity if ...
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0answers
14 views

Amortized Analysis Problem

I need to find the cost of the problem below but I cannot find a pattern to find the total cost. Problem: The cost of multiplication of n-bit binary number by 2. If a bit changes the cost of that ...
1
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1answer
43 views

$\lim\limits_{n \to \infty} \sup_{k \geq n} (\frac{1+a_{k+1}}{a_k})^k \ge e$ for any sequence $\{a_n\}$ with positive terms

Show that $$\lim_{n \to \infty} \sup_{k \ge n} \left(\frac{1+a_{k+1}}{a_k}\right)^k \geq e$$ for any sequence $\{a_k\}$ with positive terms, and that this estimate cannot be improved. Let $$s_k = ...
0
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1answer
26 views

Linear transformation from $R^2$ to $R^2$.

Let $\vec{f}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, where $\vec{f} (\vec{x}) = (x+y^2, x^3+5y)$ and $\vec{x} = (x,y) \in \mathbb{R}^2$. Let $\vec{h} = (h_1, h_2)$ and $\vec{a} = (1,1) \in ...
1
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1answer
33 views

To prove that these matrices are invertible

Let $A$ and $B$ be $n \times n$ matrices such that $||I - AB|| < 1$. Prove that $A$ and $B$ are invertible, and $$A^{-1} = B \sum\limits_{k=0}^{\infty} (I - AB)^k \text{ and } B^{-1} = ...
5
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1answer
53 views

Whats the differences between the real-entire functions on $\mathbb R^{2}$ and complex entire functions on $\mathbb C$?

We note, as set of points, $\mathbb R^{2}= \mathbb C.$ A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point ...
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0answers
14 views

conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
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0answers
17 views

Cauchy integrals over a line

Can we generalize the Cauchy integral formula from a circle to a line? Since for real integrals, the following types of improper integrals do not converge, is it correct or not that for $z\notin ...
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0answers
25 views

showing a condition implies convergence to invariant distribution

For an arbitrary Markov chain, I'm trying to show that $\lVert{ P_{\mu}\circ \theta_n^{-1} - \nu \rVert}\to 0$ iff $\sum_j \lvert{ P_{ij}^n - \nu(j)\rvert}\to 0$, where $\theta_n^{-1}$ is a shift ...
2
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1answer
19 views

The pointwise limit of increasing functions on $[0,1]$ is increasing.

For each $n \in \mathbb{N}$, let $f_{n}: [0,1] \rightarrow \mathbb{R}$ be an increasing function on $[0,1]$. Suppose that $\{f_{n}\}$ converges pointwise to a continuous function $f$ on $[0,1]$. ...
1
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1answer
15 views

Showing two sets have touching boundaries (function sets)

Let $f: [0,1] \rightarrow \mathbb{R}$ be continuous on $[0,1]$ and differentiable on $(0,1)$. Suppose that $f(0)< 0 < f(1)$ and $f'(x) \neq 0$ for every $x \in (0,1)$. Let $S_{1} = \{ x \in ...
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3answers
47 views

Show a non-empty open and closed set in R must be equal to R

I did this in class, and got no credit. We are now supposed to find a proof that works, can anyone help me with this? Thanks!
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2answers
29 views

If f is continuos on an interval, is it then uniformly continuous

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I now know that it is not. Can someone give me a proof ...
1
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1answer
34 views

Show the image of a continuous function on a closed interval is closed.

I tried this problem on my own, but got 1 out of 5. Now we are supposed to find someone to help us. Here is what I did: Let $f:[a,b] \rightarrow \mathbb{R}$ be continuous on a closed interval $I$ ...
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0answers
27 views
+50

Measurability of upper and lower derivatives of Radon measures

Let $\mu$ and $\nu$ be Radon measures in $\mathbb R^N$. Define their upper and lower derivatives by $$ \overline{D}_\nu\mu(x):=\limsup_{r\to0}\frac{\mu(B_r(x))}{\nu(B_r(x))},\qquad ...
0
votes
1answer
28 views

Change of variable (or measure)?

Hi Everyone: I am reading a book and there is a kind of "change of variable" they make that I do not understand fully. This is what they do: let $B(x,r)$ be a ball of $\mathbb{R}^{N}$ $(N>1)$, ...
0
votes
1answer
23 views

If $f$ is continuous on $(0,5)$, is it uniformly continuous on same interval

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I believe it is. I now know that it is not. Can someone ...